An aspect of this invention relates to the characterization of porous powders of geological and synthetic minerals used in industrial gas separation and solid catalyzed chemical processes wherein the speed of molecular diffusion in the porous powder is measured. In porous powders, different molecules diffuse at different speeds, that is, they have different diffusion coefficients, and the difference in the diffusion coefficients mainly determine how well they perform in the separation of gases or as catalysts. The embodiments provide a simple method and apparatus for measuring a molecular diffusion coefficient in a porous powder. The term “porous powder” is to be understood to refer to powders that comprise minerals that have porous cavities or channels and also those that comprise aggregates of nonporous minerals compressed to form porous particles. The term “diffusion coefficient” is also known as the “diffusivity”.
The speed at which molecules move into and out of a porous powder, which is also known as the internal diffusion rate, can limit the reaction rate and control the selectivity when the powder is the catalyst of a reaction. Knowledge of this diffusion rate can be used to guide the manufacture of a better catalyst. In another important application, differences in the diffusion rates can be utilized to separate the components in a gas mixture. In this invention, the term “diffusion” is used in a narrower sense than in general to mean the internal diffusion of molecules inside a porous powder. Depending on the size of the pores in the powder, the molecules would diffuse as molecules in a gas phase or when the pores are so small as to be of molecular dimensions and there is no distinct gas phase, they diffuse as an adsorbed phase. The diffusion coefficient for the molecular diffusion inside a porous powder is the parameter which is used to characterize the diffusion rate. Thus, the ability to easily carry out the quantitative measurement of the diffusion coefficient is highly advantageous for the development of useful porous powders.
The techniques for the measurement of a diffusion coefficient are described in the monograph, Diffusion in Nanoporous Materials, by Jorg Karger, Douglas M. Ruthven, and Doros N. Theodorou (Wiley-VCH, 2012), who divided these techniques into microscopic and macroscopic techniques. The microscopic techniques are based on the random walk or Einstein description of diffusion and they use the tracing of the path of tracer molecules inside the pores to measure the diffusion coefficient. However, because these techniques must trace particular molecules, they must be capable of the detection of distinct molecules and their molecular movement, and in general, they use expensive instrumentation and sophisticated procedures, e.g., the use of pulsed field gradient nuclear magnetic resonance, quasi-elastic neutron scattering, interference microscopy or confocal fluorescence microscopy. These techniques are not suitable for an ordinary research laboratory or for industrial use.
In a common research or industrial laboratory, the use of one of the macroscopic techniques is more reasonable. These have been based on sorption kinetics and a relaxation method which is usually a step response method, where a step change from P0 to P∞ is made in the gas pressure of the gas surrounding the porous powder and the curve of the concentration of the molecules in the porous powder versus time (called the soption or uptake curve) is measured as the system relaxes to its new equilibrium state. In the prior art, most experimental techniques measure the uptake curve by using a highly sensitive microbalance to measure the change in the weight of the powder versus time. However, this is disadvantageous because these microbalances are very expensive, and so there is an incentive to instead measure some changes in the gas surrounding the powder versus time as the response curve from which the uptake curve can be deduced, e.g., in the prior art, the gas replenishment rate is used, but so far there has been no easy technique for this measurement. Such a response curve is also usually known as an uptake curve, so the method is also known as the uptake curve method. The diffusion coefficient is obtained by a curve fitting method with the use of Fick's Second Law to describe the diffusion in the porous powder, namely, Eq. (1)
                                          ∂            q                                ∂            t                          =                              D            c                    ⁢                                                    ∂                2                            ⁢              q                                      ∂                              x                2                                                                        (        1        )            and its initial and boundary conditions
                              t          <          0                ,                                  ⁢                  P          =                      P            0                          ,                                  ⁢                  q          =                                    q              0                        ⁡                          (                              ∀                x                            )                                                          (                  2          ⁢          a                )                                          t          ≥          0                ,                                  ⁢                  P          =                      P            ∞                          ,                                  ⁢                              q            ⁢                          ❘                              x                =                R                                              =                      q            ∞                          ,                                  ⁢                                                            ∂                q                                            ∂                x                                      ⁢                          ❘                              x                =                0                                              =          0                                    (                  2          ⁢          b                )                                                      q            0                    =                      f            ⁡                          (                              P                0                            )                                      ,                                  ⁢                              q            ∞                    =                      f            ⁡                          (                              P                0                            )                                                          (                  2          ⁢          c                )            In Eqs. 1 and 2, P is the pressure, q is the molecular concentration inside the porous powder particle, t is the time variable, x is the space variable, Dc is the diffusion coefficient parameter, the subscript 0 denotes initial condition, the subscript ∞ denotes the new equilibrium condition, R is the length of the diffusion pathway in the particle, and q=f(P) denotes the functional form of the equilibrium relationship between the concentration inside the particle and the gas pressure, which is also known as the adsorption isotherm. Briefly, Eqs. 1 and 2 are used with an assumed value of the diffusion coefficient parameter to compute a simulated uptake curve which is compared with the experimental uptake curve. The value of the diffusion coefficient parameter in Eq. (1) is then changed to optimize the fit between the simulated and experimental uptake curves, with its value at the best fit being used as the measured diffusion coefficient.
In the prior art, the methods and apparatuses use an analytical solution of Eqs. 1 and 2 to compute the simulated uptake curve. Since an analytical solution is only available when the boundary condition which comprises the pressure of the gas environment surrounding the particle is a constant pressure, namely, Eq. 2b, therefore, the measurement has to be made with a constant pressure of gas. That is, in these methods and apparatuses, the gas environment of the particle has to be held at a constant pressure. Thus, when a step response method is used, the change in the gas environment was a very rapid change from one constant pressure to another constant pressure. In most of the prior art, the second pressure is maintained constant by using a very large volume of gas environment surrounding the powder so that the amount of gas that diffuse into the powder is a negligible fraction of it. However, this means that an auxiliary means must be used to measure the amount of gas that has diffused into the powder, and most apparatuses use a highly sensitive microbalance, which has the disadvantage that such a microbalance is very expensive and even then still does not have the very high sensitivity desired. An example of this type of apparatus was described in a paper by Youngquist, Allen and Eisenberg (Industrial and Engineering Chemistry, Product Design Development, 10 (1971) 308). Another problem that the apparatus of this type faces is that its use of a large static volume of gas that surround the powder sample gives rise to the question of whether the gas pressure is homogeneous. An apparatus to solve this latter problem made use of a new type of microbalance that can be used in a gas flow. This apparatus which used a flowing gas system and a tapered element oscillating microbalance (TEOM) was described in a paper by Zhu, Kapteijn and Moulijn (Microporous and Mesoporous Materials, 47 (2001) 157-171). However, the TEOM is a much more expensive microbalance and it is not suitable for use in a common laboratory or industrial laboratory.
Due to the disadvantage that available commercial microbalances are very expensive and yet still cannot give a really satisfactory sensitivity and resolution, there is an incentive to develop a technique to measure some change in the gas phase surrounding the powder as the response curve instead of measuring the weight of the gas added into the powder. An example of this approach is the method and apparatus in U.S. Pat. No. 4,762,010 to Borghard and Schoennagel, which used a flow controller that was capable of feeding in gas at very slow flow rates to replenish or make up for the gas that has diffused into the powder in order to keep the gas pressure constant. They used the measurement of the flow rate of the make-up gas to deduce the amount of gas that has diffused into the powder versus time or the uptake curve. This patent teaches that it is necessary to have gas feed rates that are very, very slow during the measurement because the amount of gas diffusing into the powder is very small. It is evident that this need leads to many problems, such as (1) the need to have additional complicated component parts and procedures to control the very, very slow replenishment flow rate needed to maintain a constant pressure around the powder, (2) a very slow flow rate has to be measured, which was measured by a pressure change, but the corresponding pressure change was extremely small, which made the control of the flow rate very difficult and resulted in poor accuracy, and (3) in order to make the pressure change larger, the container that supplied the feed gas was made to be very small, but because it was very small, its supply of feed gas was limited and the measurement can only be made for a very small range of pressure. Thus, the apparatus has not been much used.
U.S. Pat. No. 6,981,426 to Wang, Wei and Wang teaches a method that use fewer additional complicated component parts, but this was achieved at the expense of doing away with the automatic control of the required very slow replenishing gas flow rate. However, because there was no automatic control of the gas flow rate, the measurement procedure was made more complicated and tedious because a manual control of the gas flow rate had to be used, which is very inconvenient and labor intensive.
As discussed above, a basic difficulty that the prior art methods have to face is the control of the required very slow replenishment or make-up gas flow rate needed to maintain a constant pressure in the gas surrounding the powder sample. To avoid this difficulty, the more recent developments in the measurement of internal diffusion coefficients have turned to using apparatuses and methods that use a constant partial pressure of the measurement gas, instead of a constant total pressure, in the gas phase surrounding the powder sample so that the concentration change can be used to deduce the uptake curve. This is usually achieved with the use of a flow system and the use of a constant ratio of flow rates of the gases. However, gas flow controllers can only be reliably used for gas flow rates above 10 cm3 per minute. This together with that it is desirable to carry out a diffusion measurement in the linear adsorption isotherm regime, which requires the partial pressure of the measurement gas to be low and so its mole fraction in the gas phase should be only a few percent, requires that the total gas flow rate should be a few hundred cm3 per minute. This is a high flow rate for a laboratory apparatus, and it leads to the problem of the production of turbulence whenever there is a change in the flow rate of a component gas, such that it is not possible to produce a sharp change in the gas mole fractions of the composition gases. This turbulence means that the gas phase composition is highly erratic and its measurement is quite unreliable for some time after the step change had been made, which includes the period with much of the significant data for calculating the diffusion coefficient. One way of trying to solve this is to use the detection of the weight change in the adsorbed phase, as was done in the work cited above by Zhu, Kapteijn and Moulijn who used a flow system and a TEOM. However, as pointed out above, the TEOM is a very expensive microbalance and not suitable for common use. Another way of solving this is to use only the data collected a long time after the step change had been made, which was invented and reported in a paper by Eic and Ruthven (M. Eic, D. M. Ruthven; Zeolites, 8 (1988) 40-45) and now known as the long time limit zero length column (ZLC) method. However, this solution is unsatisfactory because it has to assume that the powder sample is completely homogeneous, that is, the long time limit data gives the same diffusion coefficient as the short time limit data, which is unlikely with most practical samples. The ZLC method has also been developed to use the complete data set but the technique is difficult and it needs special instruction by the inventors to be able to collect the data at short times after the step change, which is highly disadvantageous for common use.
Yet another way of trying to solve the problem of the turbulence that exist after a change in gas flow rate is to use some experimental parameters to characterize its effects and then use these parameters to subtract the effects of the turbulence, which was reported in a work by Guo and coworkers (Juhua Guo, Yuxin Li, Yanghuan Huang, Dezheng Wang; Journal of Nanoscience and Nanotechnology, 14(9) (2014) 6858-6864). Although their apparatus is simple and is basically that used in the chromatographic method, the preliminary work needed to characterize the turbulence is very time consuming, and hence disadvantageous. The chromatographic method, on which this work by Guo and coworkers was based, is basically a long time limit method, which can only be used with highly homogeneous sample powders.
Other methods have also been devised to solve the shortcomings discussed above, which include the frequency response method, which has been described in a work by Yasuda and Yamamoto (Y. Yasuda, A. Yamamoto; Journal of Catalysis, 93(1) (1985) 176-181) and the temporal analysis of products (TAP) method, which has been described in a work by Keipert and Baerns (O. P. Keipert, M. Baerns, Chemical Engineering Science, 53(20) (1998) 3623-3634). However, none of the methods has received much common use, and there is still a need for a simple and convenient method to measure the diffusion coefficient. The present invention meets this need. It is based on the discovery that in order to measure the diffusion coefficient in a porous powder, there is no need to make a sharp change in the gas phase environment of the powder sample, and that a gradual change can be equally well used. This resulted in a means to measure the molecular diffusion coefficient in a porous powder that used only the simple apparatus and procedure used to make a gas adsorption measurement. The measurement technique used in this invention is part of the art generally referred to as “physical modeling” in which the diffusion coefficient is measured by the optimization of the diffusion coefficient parameter in the physical model where the criterion used in the optimization is the best fit between the uptake curve calculated by the physical model and the experimentally measured uptake curve.